Finite Group Databasebeta

← #26234644

Subgroups — #26234644

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689221C211yesmaximal, characteristic
2C2×#15689221C211yesmaximal, characteristic
3C2×C4×#10625336C211yesmaximal, characteristic
4#20246299C211yesmaximal, characteristic
5#20244123C211yesmaximal, characteristic, complemented
6#20244118C211yesmaximal, characteristic, complemented
7#20243656C211yesmaximal, characteristic, complemented
8#20243648C211yesmaximal, characteristic
9#20243642C211yesmaximal, characteristic, complemented
10#20243634C211yesmaximal, characteristic
11#20243554C211yesmaximal, characteristic, complemented
12#20243403C211yesmaximal, characteristic, complemented
13#20243385C211yesmaximal, characteristic
14#20243263C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C47C211yesmaximal, characteristic, complemented
16C22×#10625336C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18C2×C4×D188C2211yescharacteristic
19C2×C4×D188C2211yescharacteristic
20#15689221—24no—
21#15689221C418yescomplemented
22C4×#10625336—24no—
23C2×#13093957C2211yescharacteristic
24C2×#13093957C2211yescharacteristic
25C2×#13093957C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×#13093955C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×#13093948C2211yescharacteristic
30C2×#13093948C2211yescharacteristic
31C2×#13093948C2211yescharacteristic
32C2×#13093944C2211yescharacteristic
33C2×#13093944C2211yescharacteristic
34C2×#13093944C2211yescharacteristic
35C2×#13093935C2211yescharacteristic
36C2×#13093935C2211yescharacteristic
37C2×C4×C47⋊C8C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689021C2211yescharacteristic
40#15688994—21no—
41#15688994—21no—
42#15688989C2211yescharacteristic
43#15688982—21no—
44#95×D47—21no—
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688878C2211yescharacteristic
50#15688878C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C47×#389—21no—
55C47×#389—21no—
56C47×#389C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#384—21no—
59C47×#384C2211yescharacteristic
60C4×C4⋊C4×C47—21no—
61C4×C4⋊C4×C47C2211yescharacteristic
62C22×D188C2311yescharacteristic
63C22×C4×D47—21no—
64C22×C4×D47—21no—
65C22×C4×D47—21no—
66C2×#10625336C2×C414yes—
67C2×#10625336D414yes—
68C2×#10625336Q814yes—
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C4×D188—22no—
74C4×D188—22no—
75C4×D188—22no—
76C4×D188—41no—
77C4×D188—41no—
78C4×D188C2×C414yes—
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×C4⋊Dic47C2311yescharacteristic
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87#13093957—22no—
88#13093957—41no—
89#13093957C2×C414yescomplemented
90#13093955—22no—
91#13093955—41no—
92C4⋊(C47⋊C8)—22no—
93#13093948—22no—
94#13093948—22no—
95#13093948C2×C414yescomplemented
96#13093944—22no—
97#13093944—22no—
98#13093944C2×C414yes—
99#13093935—22no—
100#13093935C2×C414yescomplemented
101C4×C47⋊C8—22no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C22×Q8×C47C2311yescharacteristic
107C4×Q8×C47—22no—
108C4×Q8×C47—22no—
109C4×Q8×C47—22no—
110C4×Q8×C47—41no—
111C4×Q8×C47—41no—
112C4×Q8×C47C2×C414yescomplemented
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47—21no—
119C2×C4⋊C4×C47—21no—
120C2×C4⋊C4×C47D411yescharacteristic
121C2×C4⋊C4×C47D411yescharacteristic
122C2×C4⋊C4×C47C2311yescharacteristic
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C42×C47—21no—
126C2×C42×C47—21no—
127C2×C42×C47—21no—
128C2×C42×C47C2311yescharacteristic
129C2×C42×C47C2311yescharacteristic
130C2×C42×C47C2311yescharacteristic
131#95×C47—21no—
132#95×C47—21no—
133#95×C47D411yescharacteristic
134#95×C47D411yescharacteristic
135C23×D47—21no—
136C2×Q8×C47—21no—
137C2×Q8×C47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47C22×C412yes—
140C2×Q8×C47C2×D412yes—
141C2×Q8×C47C2×Q812yes—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2411yescharacteristic
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—21no—
156C2×D188—21no—
157C2×D188—21no—
158C2×D188—21no—
159C2×D188C22×C412yes—
160C2×D188C2×D412yes—
161C2×D188C2×Q812yes—
162C2×C4×D47—24no—
163C2×C4×D47—42no—
164C2×C4×D47—42no—
165C2×C4×D47—42no—
166C2×C4×D47—42no—
167C2×C4×D47—81no—
168C2×C4×D47—81no—
169C2×C4×D47—81no—
170C2×C4×D47—81no—
171C4⋊C4×C47—24no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—41no—
179C4⋊C4×C47—42no—
180C4⋊C4×C47—42no—
181C4⋊C4×C47—42no—
182C4⋊C4×C47—81no—
183C4⋊C4×C47—81no—
184C4⋊C4×C47—81no—
185C4⋊C4×C47C22×C412yes—
186C42×C47—24no—
187C42×C47—21no—
188C42×C47—21no—
189C42×C47—21no—
190C42×C47—21no—
191C42×C47—42no—
192C42×C47—42no—
193C42×C47C22×C412yes—
194C42×C47#4112yes—
195#10625336C4⋊C4116yescomplemented
196(C2×D47)⋊C4—24no—
197(C2×D47)⋊C4—42no—
198(C2×D47)⋊C4—42no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47C22×C412yes—
205Dic47⋊C4—81no—
206Dic47⋊C4—81no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8C22×C412yes—
216C2×C47⋊C8C22×C412yes—
217C2×C47⋊C8C2×D412yes—
218C2×C47⋊C8C2×Q812yes—
219C23×C47C22×D411yescharacteristic
220C22×D47—22no—
221C22×D47—22no—
222C22×D47—22no—
223C22×D47—41no—
224C22×D47—41no—
225C22×D47—41no—
226C22×D47—41no—
227Q8×C47—22no—
228Q8×C47—22no—
229Q8×C47—22no—
230Q8×C47C2×C4⋊C414yes—
231C2×C4×C47—22no—
232C2×C4×C47—21no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—22no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—21no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47C4×D412yes—
259C2×C4×C47C4×D412yes—
260C2×C4×C47#13712yes—
261C2×C4×C47C23×C411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C22×Q811yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47H(5,𝔽2)11yescharacteristic
266C2×C4×C472−1+411yescharacteristic
267C2×Dic47—22no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276D188—22no—
277D188—22no—
278D188—22no—
279D188C2×C4⋊C414yes—
280C4×D47—84no—
281C4×D47—84no—
282C4×D47—84no—
283C47⋊C8—22no—
284C47⋊C8—41no—
285C47⋊C8—41no—
286C47⋊C8C2×C4⋊C414yes—
287#22678—471nomaximal, Sylow
288C22×C47C2×C4×D411yescharacteristic
289C22×C47D4211yescharacteristic
290C22×C47D4×Q811yescharacteristic
291C22×C47C2×#13711yescharacteristic
292C22×C47#57511yescharacteristic
293C22×C47#57611yescharacteristic
294C22×C47#57711yescharacteristic
295C2×D47—24no—
296C2×D47—42no—
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—42no—
300C2×D47—42no—
301C2×D47—42no—
302C4×C47—24no—
303C4×C47—22no—
304C4×C47—22no—
305C4×C47—22no—
306C4×C47—42no—
307C4×C47—41no—
308C4×C47—41no—
309C4×C47—41no—
310C4×C47—81no—
311C4×C47—81no—
312C4×C47C22×C4⋊C411yescharacteristic
313C4×C47#52011yescharacteristic
314C4×C47#54711yescharacteristic
315C4×C47#55411yescharacteristic
316Dic47—42no—
317Dic47—81no—
318Dic47—81no—
319C2×#440—471no—
320C2×#440—471no—
321C2×C4×SD16—471no—
322Q8×C4⋊C4D4711yescharacteristic, complemented
323#2349—471no—
324#1943—471no—
325#1942—471no—
326#1865—471no—
327#1864—471no—
328#1863—471no—
329#1862—471no—
330#1847—471no—
331#1805—471no—
332#1802—471no—
333#1771—471no—
334C2×C47D4×C4⋊C411yescharacteristic
335C2×C47C4×#13711yescharacteristic
336C2×C47#294611yescharacteristic
337C2×C47#326611yescharacteristic
338C2×C47#326811yescharacteristic
339C2×C47#338511yescharacteristic
340C2×C47#338811yescharacteristic
341D47—44no—
342C22×SD16—471no—
343C2×C4×Q8C2×D4711yescharacteristic
344C2×C4×Q8C2×D4711yescharacteristic
345C2×C4×Q8C2×D4711yescharacteristic
346C2×C4×D4—471no—
347C2×C4×D4—471no—
348C2×C4×D4—471no—
349#440—478no—
350#440—944no—
351C4×SD16—944no—
352C2×#107—471no—
353C2×#107—471no—
354C2×#107—471no—
355C2×#106—471no—
356C2×#106—471no—
357C2×C4⋊C8—471no—
358C2×#103—471no—
359C2×#103—471no—
360C2×#103—471no—
361C2×#102—471no—
362C2×#102—471no—
363C2×#102—471no—
364C2×#97—471no—
365C2×#97—471no—
366C2×C4×C8—471no—
367#389—21no—
368#389—21no—
369#389—471no—
370#389—471no—
371#389C2×D4711yescharacteristic
372#389C2×D4711yescharacteristic
373#385—941no—
374#385—941no—
375#384—21no—
376#384C2×D4711yescharacteristic
377#382—471no—
378#380—941no—
379C4×C4⋊C4—21no—
380C4×C4⋊C4C2×D4711yescharacteristic
381C2×#95—941no—
382#340—471no—
383#340—471no—
384#340—471no—
385#340—471no—
386#336—471no—
387#336—471no—
388C47#2267811yescharacteristic, complemented, Sylow
389C22×Q8C22×D4711yescharacteristic
390C22×D4—471no—
391C23×C4—941no—
392C23×C4—941no—
393C23×C4—941no—
394C2×SD16—474no—
395C2×SD16—474no—
396C2×SD16—474no—
397C22×C8—471no—
398C22×C8—471no—
399C22×C8—471no—
400C22×C8—471no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—41no—
405C4×Q8—41no—
406C4×Q8C4×D4714yescomplemented
407C4×D4—474no—
408C4×D4—942no—
409C4×D4—942no—
410C4×D4—942no—
411C4×D4—1881no—
412C4×D4—1881no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—471no—
423C2×C4⋊C4—941no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4C22⋊D4711yescharacteristic
426C2×C4⋊C4C22⋊D4711yescharacteristic
427C2×C4⋊C4C22×D4711yescharacteristic
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—941no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D4711yescharacteristic
437C2×C42C22×D4711yescharacteristic
438C2×C42C22×D4711yescharacteristic
439#107—474no—
440#107—942no—
441#107—1881no—
442#106—942no—
443#106—1881no—
444C4⋊C8—942no—
445#103—474no—
446#103—942no—
447#103—942no—
448#102—474no—
449#102—942no—
450#102—942no—
451#97—474no—
452#97—942no—
453C4×C8—942no—
454#95—21no—
455#95—21no—
456#95—941no—
457#95—941no—
458#95—941no—
459#95—941no—
460#95C22⋊D4711yescharacteristic
461#95C22⋊D4711yescharacteristic
462C24—941no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×C4×D4712yes—
467C2×Q8D4×D4712yes—
468C2×Q8Q8×D4712yes—
469C2×D4—472no—
470C2×D4—472no—
471C2×D4—472no—
472C2×D4—941no—
473C2×D4—941no—
474C2×D4—941no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—941no—
480C22×C4—944no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4—3761no—
484C22×C4—3761no—
485C22×C4—3761no—
486C22×C4—3761no—
487C22×C4—1882no—
488C22×C4—1882no—
489C22×C4—1882no—
490C22×C4—1882no—
491C22×C4C2×C22⋊D4711yescharacteristic
492C22×C4C2×C22⋊D4711yescharacteristic
493C22×C4C2×C22⋊D4711yescharacteristic
494C22×C4C2×C22⋊D4711yescharacteristic
495C22×C4C2×C22⋊D4711yescharacteristic
496C22×C4C2×C22⋊D4711yescharacteristic
497C22×C4C23×D4711yescharacteristic
498SD16—4716no—
499C2×C8—472no—
500C2×C8—472no—
501C2×C8—472no—
502C2×C8—472no—
503C2×C8—941no—
504C2×C8—941no—
505C2×C8—941no—
506C2×C8—941no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C2×C8—941no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—41no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—472no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—941no—
531C4⋊C4—3761no—
532C4⋊C4—3761no—
533C4⋊C4C2×C4×D4712yes—
534C22⋊C4—944no—
535C22⋊C4—1882no—
536C22⋊C4—1882no—
537C42—24no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—42no—
543C42—42no—
544C42C2×C4×D4712yes—
545C42#1062535012yes—
546C23—942no—
547C23—942no—
548C23—942no—
549C23—1881no—
550C23—1881no—
551C23—1881no—
552C23—1881no—
553C23C22×C22⋊D4711yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8C4⋊C4×D4714yes—
558D4—474no—
559D4—942no—
560D4—942no—
561D4—942no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—942no—
590C2×C4—3764no—
591C2×C4—3764no—
592C2×C4—3764no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4C4×C22⋊D4712yes—
602C2×C4C4×C22⋊D4712yes—
603C2×C4#1309405512yes—
604C2×C4C22×C4×D4711yescharacteristic
605C2×C4C2×D4×D4711yescharacteristic
606C2×C4C2×Q8×D4711yescharacteristic
607C2×C4C2×#1062535011yescharacteristic
608C2×C4#1309406911yescharacteristic
609C2×C4#1309407211yescharacteristic
610C8—474no—
611C8—942no—
612C8—1881no—
613C8—1881no—
614C22—944no—
615C22—1882no—
616C22—1882no—
617C22—1882no—
618C22—1882no—
619C22—1882no—
620C22—1882no—
621C22C2×C4×C22⋊D4711yescharacteristic
622C22D4×C22⋊D4711yescharacteristic
623C22Q8×C22⋊D4711yescharacteristic
624C22C2×#1309405511yescharacteristic
625C22#1568992111yescharacteristic
626C22#1568992611yescharacteristic
627C22#1568993111yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—81no—
637C4—81no—
638C4—3761no—
639C4—3761no—
640C4—1882no—
641C4C2×C4⋊C4×D4711yescharacteristic
642C4#1568960411yescharacteristic
643C4#1568977411yescharacteristic
644C4#1568982811yescharacteristic
645C2—1884no—
646C2C4⋊C4×C22⋊D4711yescharacteristic
647C2C4×#1309405511yescharacteristic
648C2#2025221611yescharacteristic
649C2#2025489911yescharacteristic
650C2#2025492911yescharacteristic
651C2#2025629411yescharacteristic
652C2#2025632711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingQ8×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenterQ8×C4⋊C4D47
hypocenterC47#22678
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #20244123C2semidirect
6. #20244118C2semidirect
7. #20243656C2semidirect
9. #20243642C2semidirect
11. #20243554C2semidirect
12. #20243403C2semidirect
14. #20243263C2semidirect
15. Q8×C4⋊C4×C47C2semidirect
21. #15689221C4semidirect
89. #13093957C2×C4semidirect
95. #13093948C2×C4semidirect
100. #13093935C2×C4semidirect
112. C4×Q8×C47C2×C4semidirect
195. #10625336C4⋊C4semidirect
322. Q8×C4⋊C4D47semidirect
388. C47#22678semidirect
406. C4×Q8C4×D47semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156892212
C2×C4×#106253362
#202462992
#202441232
#202441182
#202436562
#202436482
#202436422
#202436342
#202435542
#202434032
#202433852
#202432632
Q8×C4⋊C4×C472
#2267847